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Why studying number theory? This is my question for the whole week. I very interested in studying number and it's properties, recently I started asking myself this question, why number theory?

• Because professor says you must.

• Because you won’t graduate if you don’t.

• Because you have to take something.

• Because it gives your mind valuable training in thinking logically.

• Because numbers might be interesting.

• Because numbers are a fundamental part of a person’s mental universe and hence worth looking into.

• Because some of the most powerful human minds that ever existed were concerned with numbers, and what powerful minds study is worth studying.

• Because you want to know everything about numbers: what makes them work and what they do.

• Because mathematics contains some beautiful things, and someone told you that number theory contained some of the most beautiful – and few of the most ugly – things.

• Because it is fun.

• Because there are still many easy-to-state unsolved problems in number theory, and that’s cool (and driving a lot of powerful minds crazy).


Well, what is my answer? Can you help me to select from the above?

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I don't know why you study number theory. Do you study number theory? I specialize in number theory because I like it, and that's enough for me. –  mixedmath Mar 7 '12 at 3:41
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....... Why not? –  01000100 Mar 7 '12 at 3:45
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@Hassan: I think this is a very 'deep' question. Why do you do anything? Some people do not get a sense of accomplishment by merely doing things that they like. I don't know if you do/would. From time to time, I wrestle with that myself. –  mixedmath Mar 7 '12 at 3:57
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Number theory is off topic on MSE? OP supplies many possible answers and ask help to make selection. How much better we want him to do? If you don't like any of the possible answers, give him another one! That's all he asked. –  scaaahu Mar 7 '12 at 4:25
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"Off topic" was probably a poor choice. "Not constructive" would have been better (The system does not let you undo mistakes like this, sadly). I couldn't see what this question was going to accomplish. Is it a survey? Is it crowd-sourced soul-searching? Those are both important endeavors but this doesn't seem like the place for them. –  Dylan Moreland Mar 7 '12 at 4:45
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closed as off topic by Dylan Moreland, mixedmath, Byron Schmuland, Austin Mohr, Zev Chonoles Mar 7 '12 at 4:06

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4 Answers

I must admit myself that this is a great question. At first I thought number theory was quite useless ; not because its theorems were not useful at all or whatever, but because when one studies advanced number theory, he realizes that most questions asked are not pertinent in their solution but more in the argument(s) used to go through to the solution ; the more complicated / sophisticated those arguments are, the better feeling we get out of it.

I feel one reason to study analysis is optimization, because things need to be optimized in this world. One reason to study algebra is to understand better the structures in this world. One reason to study geometry is to see how figures in the plane/space/other weird things behave around us. Those reasons felt intuitive to me and I thought it was enough to drive someone into research in one of those fields.

But number theory? If you'd ask only me, I think it started by being central in mathematics (speaking of 2000 years ago and such), and then mathematics got diversified, but somehow number theory pops up every now and then, until we come to today, where number theory is present everywhere and numbers keep putting bridges between all the fields in mathematics.

My research director is a number theorist, and I am never bored when I attend talks from his colleagues ; algebra, analysis, geometry, topology, probability, you see all of it when you study number theory. I don't like analytic number theory a lot, and at first I thought that was all of it, so I gave a quick glance at it and said "bleh, I hate it". When I discovered algebraic number theory (at which I am still a newbie), I suddenly "liked" number theory (at least that part). Number theory is a very vague subject that will keep popping back in every field of mathematics you choose ; depending on the field it will pop out more or less, but you see my point.

One example that I like is in representation theory : when decomposing the $FG$-module $FG$ as a direct sum of irreducible $FG$-modules using Maschke's Theorem, one realizes that the dimension of $FG$ is a sum of squares. Counting the possibilities for the dimension gives us the possibility for the degree of the representation and the size of the blocks.

Hope that helps,

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Because $x^n+y^n=z^n$ seems not to have any positive integer solutions for $n>2$, and while I can verify this by computer up to huge numbers, I see no reason why this should be true in general. How can that not captivate me? Of course, Fermat hand an elementary proof but failed to write it down...

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Is $x^n+y^n=z^n$ insolvable for $n>2$ only your reason for studying number theory? –  Hassan Muhammad Mar 7 '12 at 4:06
    
@HassanMuhammad No, it's simply an example. –  Alex Becker Mar 7 '12 at 4:11
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One of its uses is cryptology, here is more detail: http://www.enotes.com/cryptology-number-theory-reference/cryptology-number-theory Also, it have application in computational mathmatics, an example would be: http://en.wikipedia.org/wiki/Bailey-Borwein-Plouffe_formula

that is to compute pi in base two

By the way, number theory also study the rational numbers properties other than the property of integer, which is also very important

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I think you understand me. I asked why studying number, is it because of there application in nature? –  Hassan Muhammad Mar 7 '12 at 4:02
    
@HassanMuhammad - Of course, That is why i list some of the applications, it is fun for learning number theory, isn't it? –  Victor Mar 7 '12 at 4:04
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Number theory is natural as a branch of mathematics, since historically math began by studying basic properties of numbers. Far from being removed, the theory appears in many aspects of mathematics, from practical applications like cryptography to more abstract realms.

I personally am interested in topology, since I tend to think more with images and spatial sense. I used to think "Who the heck cares about number theory?" As I began learning the algebraic side, I discovered that you can't live without knowing at least some of it! The cohomology ring of a topological space is a stronger invariant than homology precisely because it is endowed with a ring structure. As I began learning about geometric topology I began noticing Bernoulli numbers in places like Hirzebruch's signature theorem and in the order of the group of homotopy $n$-spheres (under connected sum, up to h-cobordism).

The Bernoulli numbers themselves have a strange tendency to appear everywhere in mathematics, which is just a suggestion of number theory's far-reaching effects.

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